Questions tagged [complex-manifolds]

For questions about complex manifolds.

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10 views

Are the two parts of a Hermitian metric compatible in the Levi-Civita sense?

Consider a Hermitian metric $h$ on a complex vector bundle. We know that we can form a Riemannian metric $g$ and closed $(1,1)$ curvature form $\omega$ such that $$ h = g -i\omega $$ where $$ g = \...
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2answers
47 views

the simplest non-trivial line bundle over Riemann sphere

We define Riemann sphere as $S=\mathbb{C}^2-\{0\}/\sim$. Given a point $p$ over $S$, I have seen somewhere there exists a line bundle $L_p$ associated to $p$, and $L_p$ has a non-zero holomorphic ...
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7 views

Del-bar commuting with averaging operator

Let $G$ be a compact group (possibly finite) acting holomorphically on a complex manifold $(M, J)$. Then we have an averaging map $$\rho: \Omega^{p,q}(M) \rightarrow \Omega^{p,q}(M)$$ $$\omega \...
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1answer
21 views

Fundamental form of almost complex manifold is $(1,1)$-form

Let $M$ be an almost complex manifold with almost complex structure $J$, a compatible Riemannian metric $g$ and fundamental form $\omega$. Consider the eigenspaces $T_p^{1,0}M=\{v\in T_pM\otimes\...
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14 views

how does a complex structure on a real vector space give rise to a hodge structure

Let $V$ be a finite dimensional vector space over $\mathbb{R}$, with a complex structure, i.e. a $\mathbb{R}$-linear map $J:V \to V$ such that $J^2=-Id$ (which gives $V$ the structure of a $\mathbb{C}$...
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5 views

holomorphic section of positive line bundle

I read the following statement from the book "L^2 approaches in several complex variables" page 206: Positive dimensional analytic sets must intersect with the zeros of holomorphic sections of ...
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12 views

Conclusion of implicit function theorem in local lie group

My teacher defines local lie group as follows: Definition: A complex n-dimensional local Lie group $G$ in the neighborhood $V ⊂ \mathbb{C^n}$ is determined by a function $\phi:\mathbb{C^n} ×\mathbb{C^...
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24 views

Every closed orientable surface is Riemann surface

I want to prove that every closed orientable surface is a Riemann surface i.e. every closed orientable surface admits a complex structure. Several proofs are available which make use of classification ...
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20 views

lower bound the distance between two varieties

$\DeclareMathOperator{\complex}{\mathbb{C}}$ Let $X,Y \subseteq \complex^n$ be homogeneous, smooth, irreducible, closed algebraic sets with $X \cap Y=\{0\}$. I would like to numerically lower bound ...
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1answer
19 views

Holomorphic functions on Riemann surfaces with boundary

Suppose that $\Sigma$ is a compact Riemann surface with boundary and that $f: \Sigma \rightarrow \mathbb{C}$ is holomorphic*. If $f$ is real-valued along $\partial \Sigma$, is it necessarily true that ...
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2answers
153 views

Why restrict to complex Lie algebras?

I am taking a class about Lie algebras, where we introduced in the beginning the notion of a Lie algebra, but over time we restricted ourselves only to complex Lie algebras. Can someone of you tell me ...
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2answers
65 views

If $X$ is closed then $JX$ is closed

Let $(S, g, J)$ be a closed Riemann surface with a Riemannian metric $g$ compatible with the complex structure $J$. Suppose that a smooth vector field $X$ on $S$ is closed, i.e., the $1$-form $\omega =...
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1answer
57 views

The Projection to a Projective Line is Holomorphic

I have a line $L\subset \mathbb CP^2$ and a point $R\in \mathbb CP^2-L$. I need to prove that the map \begin{align*} \varphi:\mathbb CP^2&-\{R\} \to L\\ &P\quad\mapsto L\cap L_{RP} \end{align*...
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1answer
22 views

How is the Kähler form decomposed in terms of the metric?

In countless textbooks and lecture notes (e.g. eqn 4.9 of Lectures on Riemannian Geometry, Part II: Complex Manifolds by Stefan Vandoren), the Kähler (1,1)-form, $\omega$, is written in terms of the ...
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9 views

The integrability of almost complex structure in the sense of Frobenius theorem.

I've tried to distinguish the almost complex structure and complex structure, intuitively. Without any chance of confusion, I'll assume that the manifolds and the maps are "smooth". The almost ...
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30 views

what's the diffrence between an holomorphic and a real plane?

I am studying holomorphic sectional curvature, and I see that the diffrence between it and an real sectional curvature is that one is the restriction of the sectional curvature to an holomorphic ...
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22 views

Elliptic equation in complex variables

Given a complex manifold $M$ equipped with a hermitian metric $g$, one can define a Laplace operator on it by $\Delta u = g^{i \bar j} \partial_i \partial_{\bar j} u$. The claim is that in real ...
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1answer
78 views

When does the pullback of differential forms preserve cohomology classes: $[\alpha]=[F^* \alpha]$?

Let $M^n$ be a smooth manifold, compact with no boundary. Consider a diffeomorphism $F: M \rightarrow M.$ The pullback $F^*$ defines a linear isomorphism on the de Rham cohomogy spaces $F^*: H^k_{dR}(...
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1answer
61 views

Showing that $\tau(t) = (t^2, t^3)$ is not a submanifold

Let $\tau : \mathbb{C} \to \mathbb{C}^2$ be the map $\tau(t) := (t^2, t^3)$. Show that $\tau$ defines an embedding map from $\mathbb{C}^*$ to $\mathbb{C}^2 \setminus{0}$. Is $\tau(\mathbb{C})$ a ...
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1answer
27 views

Are the chart maps of a complex manifold necessarily biholomorphic?

I know that the transition maps of a complex manifold are biholomorphic, but are the chart maps themselves also biholomorphic? I know that it is the case for real smooth manifolds (here the chart maps ...
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35 views

Dimension of the complex projective space $\mathbb{C}\mathbb{P}^n$ as almost complex manifold

I have already shown that the complex projective space $\mathbb{C}\mathbb{P}^n$ is a complex manifold by checking the required properties of the transition maps. Since every complex manifold is an ...
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16 views

Holomorphic function spaces on Stein manifolds

This may be a long shot, but I'm interested in learning about holomorphic function spaces on Stein manifolds, however, I can't find much literature on the topic. I don't know if this is just too hard ...
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1answer
31 views

Does uniformization theorem imply all 2d manifolds are confromally flat?

Here is a screenshot from nakahara. Now, to me it looks like that the uniformization theorem implies that all 2d manifolds are conformally flat, because the constant curvature metrics described in (14....
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1answer
61 views

complex manifolds and geometry reference request

I am curious to learn about complex manifolds and complex geometry. I am familiar with the classical algebraic and analytic theory of Riemann surfaces, complex analysis in one variable (say, first ...
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2answers
99 views

Manifold and hyperplane

Can someone explain me the relation between manifold and hyperplane. I saw a definition but I am not able to connect the idea. The definition is A set Γ ⊂ $R^n $ is called a k–dimensional $C_m$...
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51 views

Singularities of moduli spaces $M_g$

This survey paper from Lizhen Ji says: Teichmüller was aware that nontrivial automorphisms of Riemann surfaces caused difficulty in constructing $M_g$ and singularities of $M_g$, and he ...
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28 views

Definition of bubbles and Removal of singularities

In my lecture, I have the following theorem: Suppose $u:(B^2 \setminus \{0\}, i) \rightarrow (M,J)$ is $J$-holomorphic with $E(u)< \infty$ (energy) and such that the image of $u$ is contained in ...
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1answer
35 views

CR differential operators

I don't quite understand the following definition: Let $(E,J)$ be a complex manifold, $(\Sigma, j)$ Riemann surface with a.c. structure, $E \rightarrow \Sigma$ a holomorphic bundle. Define $\...
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1answer
81 views

Definition of Weinstein manifold

I have the following definition for Weinstein manifolds in my lecture: A Weinstein manifold is an exact symplectic manifold $(W, \omega= d \lambda)$, whose associated Liouville vector field is ...
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1answer
138 views

Manifolds which are not realized with the regular value theorem

Are there smooth/holomorphic manifolds which cannot be defined using the regular value theorem? That is, they are not the preimage of a regular value?
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30 views

Kummer suface ; cohomology of the resolution

I have questions regarded to the resolution of Kummer surface. You can see the other 2 ones here At first, I am describing a resolution of Kummer surface: Get a lattice of rank 4 ; $\Gamma$ on $ \...
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35 views

Kummer suface ; the resolution has a holomorhic (2,0)-form .

At first, I am describing a resolution of Kummer surface: Get a lattice of rank 4 ; $\Gamma$ on $ \mathbb{C}^2$. The quotient $\mathbb{T}^4:\mathbb{C}^2 /\Gamma$ would be a complex tori . Now ...
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0answers
34 views

Line bundles on projective space and disk

I'm having a difficult time solving some exercice. I should prove the following : Show that any holomorphic line bundle on a disc $\Delta\subset\mathbb{C}$ is trivial. Deduce that any holomorphic ...
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1answer
45 views

Show that $\log|f|$ is a plurisubharmonic function

$\Omega \subset C^n$. $f \in O(\Omega)$. Show that $\log|f|$ is a plurisubharmonic function. I have tried two methods. The first one is calculating Hessain matrix of $log|f|$, but it is too hard. ...
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33 views

Show that the closure of $\cup_{k\geq1}F_k(D)$ is compact in $\Omega$

$\Omega$ is a domain of holomorphy, $D=\{z \in C : |z|\leq1\}$.For an arbitrary series of holomorphic functions $F_k:D\to \Omega$,The closure of $\cup_{k\geq1}F_k(\partial D)$ is compact in $\Omega$. ...
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2answers
66 views

Fibre of smooth holomorphic map is manifold (ComGeo by Huybrechts)

I have a question on a remark from Daniel Hyubrechts' Complex Geometry Complex Geometry on page 107. Definition 2.6.13 A holomorphic map $f: X \to Y$ is smooth at a point $x \in X$ if the induced ...
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50 views

Equivalent definitions of tangent spaces (complex VS real)

$\newcommand{\CA}{{\mathcal{A}}} \newcommand{\CG}{{\mathcal{G}}} \newcommand{\BR}{{\mathbb{R}}} \newcommand{\Fm}{{\mathfrak{m}}} \newcommand{\smint}{{(-\varepsilon,\varepsilon)}} \newcommand{\seq}{{\...
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41 views

Equivalence between etale, smooth, and unramified morphisms and local diffeomorphisms, submersions, and immersions

It is known that if a morphism between two smooth quasiprojective complex algebraic varieties is etale at a point, then it is a local diffeomorphism in a neighborhood of that point. Similarly, does a ...
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2answers
125 views

Path to Manifolds from HS Algebra and Calculus?

Is there a coherent path from high school algebra and beginning calculus to fully understanding the manifolds? In other words, can one self-study towards manifolds, only assuming a very modest ...
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1answer
36 views

How to see if a (1,1)-form stems from a hermitian metric

In Griffiths & Harris Principles of Algebraic Geometry on page 28 it is explained how you derive a real differential (1,1)-form $\omega$ from a hermitian metric $ds^2$ on a complex monifold $M$. ...
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1answer
92 views

Complexified Tangent Bundle of a Riemann Surface

As we know the Complexified Tangent Space of a Riemann Surface $M$ at a point $p$ is $T_{\mathbb{R},p}(M)\otimes_\mathbb{R}\mathbb{C}=\mathbb{C}\{\frac{\partial}{\partial x},\frac{\partial}{\partial ...
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21 views

How Quintic 3-fold is a Calabi–Yau manifold and has non vanishing Ricci scalar?

It’s well known that quintic 3-fold is a Calabi-Yau manifold in the complex projective space $\mathbb{CP}^{n+1}$ , see for instance: https://en.m.wikipedia.org/wiki/Quintic_threefold Now the main ...
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23 views

Complex manifolds vs Riemann domains

In Hörmander's text an "Introduction to Several Complex Variables," he gives the following definition for a Riemann domain on page 139: A complex manifold $\Omega$ of dimension $n$ is called a ...
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32 views

Sufficient Conditions for Deformations of a Complex Manifold to form an Almost-Quaternionic Structure

Let $(M, J)$ be a $2n$ (complex) dimensional manifold with complex structure $J$, and consider a differentiable family $\mathcal{F}=(\mathcal{M}, B, \phi)$ of complex structures with respect to $M$, ...
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1answer
38 views

Question about the linear system of divisors

This is from Griffith Harris, p.137: Let $D$ be a divisor on $M$. If $M$ is compact, for every $D'\in |D|$. There exists $f\in L(D)$ such that $D'= D + (f)$, and conversely any two such functions $f, ...
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1answer
44 views

How to prove a functional equation has finite dimensional solution space?

Consider a functional equation for meromorphic functions $S(z)$, such as the following: $$S(z^2) + w S(wz^2) + w^2 S(w^2z^2) = 3zS(z^3)-3zS(z^6),$$ where $w=e^{i2\pi/3}$. It is obvious that all ...
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17 views

Show that exists a linear connection $D$ such that $D''=d''$

I need an help, I want to understand this theorem. Statement: Let $E$ be a a holomorphic vector bundle over a complex manifold $M$. There exists a connection $D$ such that $D''=d''$ Proof: Let $U=...
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22 views

What does parametric disc mean in this context of triangulation?

Suppose I have two compact Riemann surfaces $\mathfrak{B}$ and $\mathfrak{B}'$ as well as a covering $f: \mathfrak{B}' \rightarrow \mathfrak{B}.$ My book says that I can triangulate the Riemann ...
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33 views

component of the covariant derivative is a tensor

Let $M$ be a complex manifold with a Kahler metric $g$, define the covariant derivative of a smooth complex valued $T^{(1,0)}M$ vector field $X = X^i\partial_i$ to be such that its i^{th} component is ...
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1answer
67 views

Why does the dual of a vector bundle use the inverse transpose?

I would expect that the dual of a vector bundle would be defined by the inverse conjugate transpose, as that would be the inverse of the adjoint. When $\alpha_{ij}:X\to Y$ is a transition matrix in $E$...

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